How do i find the equivalent voltage source?

[An1MuS]

Homework Statement

How do i find the equivalent voltage source of...

1) Two AC voltage sources in series with same frequency and phase = ?
2) One AC voltage source in series with a DC voltage source = ?

Homework Equations

Equation of a AC voltage source as a function of time [tex]V(t)=V_m * sin( \omega *t + \theta )[/tex]

The Attempt at a Solution

1) [tex]V(t)=(V_{m_1}+V_{m_2}) * sin( \omega *t + \theta )[/tex]
2) [tex]V(t)=(V_{DC}+V_m) * sin( \omega *t + \theta )[/tex]

 

[gneill]

Homework Statement

How do i find the equivalent voltage source of...

1) Two AC voltage sources in series with same frequency and phase = ?
2) One AC voltage source in series with a DC voltage source = ?

Homework Equations

Equation of a AC voltage source as a function of time [tex]V(t)=V_m * sin( \omega *t + \theta )[/tex]

The Attempt at a Solution

1) [tex]V(t)=(V_{m_1}+V_{m_2}) * sin( \omega *t + \theta )[/tex]
2) [tex]V(t)=(V_{DC}+V_m) * sin( \omega *t + \theta )[/tex]

Your first answer is correct. The second is not; a DC source is a constant value that does not vary sinusoidally. Suppose you had the two functions f(t) = 4 and g(t) = sin(ωt). What would be the result of adding them: f(t) + g(t) = ?

 

[An1MuS]

since f(t) = 4, then adding g(t) + f(t) would be the same as g(t) + 4...

Ah that makes sense, if i add a constant function such as DC current to some other non-constant like the sinusoidal function of AC current, it's like adding it's value, so the answer to b) is

[tex]V(t)=V_{DC}+V_m*sin(\omega *t + \theta)[/tex]

 

[gneill]

since f(t) = 4, then adding g(t) + f(t) would be the same as g(t) + 4...

Ah that makes sense, if i add a constant function such as DC current to some other non-constant like the sinusoidal function of AC current, it's like adding it's value, so the answer to b) is

[tex]V(t)=V_{DC}+V_m*sin(\omega *t + \theta)[/tex]

Much better!

 

[DeeNos]

For both of these situations, the equivalent voltage source can be found by using the superposition principle. This principle states that the total voltage in a circuit is equal to the sum of individual voltage sources.

1) For two AC voltage sources in series with the same frequency and phase, the equivalent voltage source can be found by simply adding the individual voltages. So, the equivalent voltage source would be V(t) = (V_m1 + V_m2) * sin( \omega *t + \theta ).

2) For one AC voltage source in series with a DC voltage source, the equivalent voltage source can be found by considering the DC voltage source as a constant voltage. So, the equivalent voltage source would be V(t) = (V_DC + V_m) * sin( \omega *t + \theta ). This is because the DC voltage does not vary with time, so it can be added to the AC voltage as a constant value.